There are several well known mathematical modeling techniques for estimating the risk of a portfolio of financial assets such as securities and for deciding how to strategically invest a fixed amount of wealth given a large number of financial assets in which to potentially invest.
For example, mutual funds often estimate the active risk associated with a managed portfolio of securities, where the active risk is the risk associated with portfolio allocations that differ from a benchmark portfolio. Often, a mutual fund manager is given a “risk budget”, which defines the maximum allowable active risk that he or she can accept when constructing a managed portfolio. Active risk is also sometimes called portfolio tracking error. Portfolio managers may also use numerical estimates of risk as a component of performance contribution, performance attribution, or return attribution, as well as, other ex-ante and ex-post portfolio analyses. See for example, R. Litterman, Modern Investment Management: An Equilibrium Approach, John Wiley and Sons, Inc., Hoboken, N.J., 2003 (Litterman), which gives detailed descriptions of how these analyses make use of numerical estimates of risk and which is incorporated by reference herein in its entirety.
Another use of numerically estimated risk is for optimal portfolio construction. One example of this is mean-variance portfolio optimization as described by H. Markowitz, “Portfolio Selection”, Journal of Finance 7(1), pp. 77-91, 1952 which is incorporated by reference herein in its entirety. In mean-variance optimization, a portfolio is constructed that minimizes the risk of the portfolio while achieving a minimum acceptable level of return. Alternatively, the level of return is maximized subject to a maximum allowable portfolio risk. The family of portfolio solutions solving these optimization problems for different values of either minimum acceptable return or maximum allowable risk is said to form an “efficient frontier”, which is often depicted graphically on a plot of risk versus return. There are numerous, well known, variations of mean-variance portfolio optimization that are used for portfolio construction. These variations include methods based on utility functions, Sharpe ratio, and value-at-risk.
Such portfolio construction procedures make use of an estimate of portfolio risk, and some make use of an estimate of portfolio return. A crucial issue for these optimization procedures is how sensitive the constructed portfolios are to changes in the estimates of risk and return. Small changes in the estimates of risk and return occur when these quantities are re-estimated at different time periods. They also occur when the raw data underlying the estimates is corrected or when the estimation method itself is modified. Mean-variance optimal portfolios are known to be sensitive to small changes in the estimated asset return, variances, and covariances. See, for example, J. D. Jobson, and B. Korkei, “Putting Markowitz Theory to Work”, Journal of Portfolio Management, Vol. 7, pp. 70-74, 1981 and R. O. Michaud, “The Markowitz Optimization Enigma: Is Optimized Optimal?”, Financial Analyst Journal, 1989, Vol. 45, pp. 31-42, 1989 and Efficient Asset Management: A Practical Guide to Stock Portfolio Optimization and Asset Allocation, Harvard Business School Press, 1998, (the two Michaud publications are hence referred to collectively as “Michaud”) all of which are incorporated by reference herein in their entirety.
A number of procedures have been proposed to alleviate this sensitivity problem. Michaud proposes using bootstrap resampling based on estimates of asset return, variance, and covariance to generate a distribution of efficient frontiers. R. Jagannathan and T. Ma, “Risk Reduction in Large Portfolios: Why Imposing the Wrong Constraints Helps”, Journal of Finance, Vol. 58, pp. 1651-1683, 2003, consider restrictions on portfolio weights as a way to improve portfolio construction. A Bayesian approach in which the input parameters are modified towards plausible values has been made popular by F. Black, and R. Litterman, “Global Portfolio Optimization”, Financial Analysts Journal, pp. 28-43, 1992.
More recently, mathematical techniques in robust optimization have been used to explicitly model and compensate for estimation error in portfolio risk and, where appropriate, return. The upside of robust portfolio optimization is that large arbitrage-like bets that are sensitive to model parameters can be avoided. The downside is that too much conservativeness leaves real opportunities unexploited.
Robust portfolios are constructed by solving a quadratic min-max problem with quadratic constraints. Technical details for solving such problems are given in A. Ben-Tal, and A. Nemirovski, “Robust Convex Optimization”, Mathematics of Operations Research, Vol. 23, pp. 769-805, 1998, which is incorporated by reference herein in its entirety. Robust optimization techniques have been applied to financial problems by M. S. Lobo, “Robust and Convex Optimization with Applications in Finance”, Stanford University dissertation, 2000, and D. Goldfarb, and G. Iyengar, “Robust Portfolio Selection Problems”, Mathematics of Operations Research, Vol. 28, pp. 1-37, 2003, both of which are incorporated by reference herein in their entirety.
From the above, it is seen that there is a recognition that it is important to consider and compensate for estimation and modeling errors in risk when analyzing and constructing financial portfolios. Although conceptually it is possible to distinguish modeling error, which is error due to specifying the model, and estimation error, which is error due to measurement and data errors, in practice, the two sources of error are indistinguishable and must be handled by the same techniques.
Suppose that there are N assets in an investment portfolio, and the weight or fraction of the available wealth invested in each asset is given by the N-dimensional column vector w. These weights may be the actual fraction of wealth invested or, alternatively, in the case of active risk, they may represent the difference in weights between a managed portfolio and a benchmark portfolio as described by Litterman. The risk of this portfolio is calculated, using standard matrix notation, asV=wTQwwhere V is the portfolio variance, a scalar quantity, and Q is an N×N positive semi-definite matrix whose elements are the variance or covariance of the asset returns.
Expected covariances of security returns are difficult to estimate. For N assets, there are N(N+1)/2 separate variances and covariances to be estimated. The number of securities that may be part of a portfolio, N, is often over a 1000, which implies that over 500,000 values must be estimated. Risk models typically cover all the assets in the asset universe, not just the assets with holdings in the portfolio, so N can be considerably larger than the number of assets in a managed or benchmark portfolio.
To obtain reliable variance or covariance estimates based on historical return data, the number of historical time periods used for estimation should be of the same order of magnitude as the number of assets, N. Often, there may be insufficient historical time periods. For example, new companies and bankrupt companies have abbreviated historical price data and companies that undergo mergers or acquisitions have non-unique historical price data. As a result, the covariances estimated from historical data can lead to matrices that are numerically ill-conditioned. Such covariance estimates are of limited value.
Factor risk models were developed, in part, to overcome these short comings. See for example, R. C. Grinold, and R. N. Kahn, Active Portfolio Management: A Quantitative Approach for Providing Superior Returns and Controlling Risk, Second Edition, McGraw-Hill, New York, 2000, which is incorporated by reference herein it its entirety, and Litterman.
Factor risk models represent the expected variances and covariances of security returns using a set of M factors, where M<<N, that are derived using statistical, fundamental, or macro-economic information or a combination of any of such types of information. Given exposures of the securities to the factors and the covariances of factor returns, the covariances of security returns can be expressed as a function of the factor exposures, the covariances of factor returns, and a “remainder”, called the specific risk of each security. Factor risk models typically have between 20 and 80 factors. Even with 80 factors and 1000 securities, the total number of values that must be estimated is just over 85,000, as opposed to over 500,000.
A substantial advantage of factor risk models is that since, by construction, M<<N factor risk models do not need as many historical time periods to estimate the covariances of factor returns and thus are much less susceptible to the ill-conditioning problems that arise when estimating the elements of Q individually. However, the fact that M<<N is also a disadvantage of factor risk models: the null-space of factor exposures is non-empty, which means that the factor risk model cannot capture certain risk information. Although all elements in the asset covariance matrix are estimated, the factor risk model cannot accurately estimate all possible variance/covariance matrices as the number of factors is much smaller than the number of assets. Furthermore, the accuracy of the model depends on the choice of factors and there are a huge number of possible choices for factors. This deficiency can be thought of as modeling error and is inherent in factor risk models. Additional inaccuracies are introduced due to errors when the parameters of the factor risk model are estimated.
This modeling error manifests itself when a factor risk model is used to estimate the risk of an investment portfolio, or to compute an investment portfolio using mean-variance optimization. In the first case, the estimated portfolio risk may be inaccurate. In the second case, the resulting portfolio may not be optimal.
Accordingly, among its several aspects, the present invention recognizes that there remains a need for systems and methods that can efficiently and effectively estimate Q by explicitly accounting for modeling and estimation error in factor risk models.
United States Patent Publication No. 2002/0123953 describes an approach in which a factor risk model may be altered by specifying an uncertainty set and confidence threshold for the factor loading or exposure matrix, B, and factor covariance matrix, Σ. However, the uncertainty set described is general and does not specifically depend on the linear algebraic properties of the exposure matrix, B. In United States Patent Publication No. 2004/0236546, the alterations to the risk model are said to be independent of the mathematical structure of B. By contrast, in the present invention, the alterations to improve the risk model specifically depend on the linear algebraic properties of the exposure matrix, such as the null space of this matrix.